Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
424 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27,30,23)(20,24)(21,25)(22,26)(28,41,44,35)(29,42,46,36)(31,37)(32,38)(33,39)(34,40)(43,48,45,47);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19,22,30)(18,23,26,27)(20,28,32,44)(21,29,34,46)(24,35,38,41)(25,36,40,42)(31,43,33,45)(37,47,39,48);(3,9)(6,12)(10,15)(13,16)(17,20,22,32)(18,24,26,38)(19,28,30,44)(21,33,34,31)(23,35,27,41)(25,39,40,37)(29,45,46,43)(36,48,42,47);(2,7)(6,12)(8,14)(13,16)(17,21,22,34)(18,25,26,40)(19,29,30,46)(20,31,32,33)(23,36,27,42)(24,37,38,39)(28,43,44,45)(35,47,41,48)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,c^b=ch,f^b=fg,d^c=dg46 conjugacy classes — view the full character table.
| maximal subgroup of | #139441, #139443, #139469, … |
| maximal normal subgroup of | #139441, #139443, #139469, … |
| maximal quotient of | #162955, #162963, #162971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10398);